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exists in F an element y such that ay = 1 (whence ya = 1, by
From these simple postulates the whole of common algebra
follows. A word or two about some of the statements may be
helpful to those who have not seen algebra for years. In II, the
statement (a  b) -f c = a + (b 4- c), called the associative law
of addition, says that if a and b are added, and to this sum is
added c, the result is the same as if a and the sum of b and c
are added. Similarly with respect to multiplication, for the
second statement in II. The third statement in II is called the
distributive law. In III a 'zero' and 'unity' are postulated; in IV,
the postulated x gives the negative of a; and the first paren-
thetical remark in V forbids 'division by zero". The demands in
Postulate I are called the commutative laws of addition and
m ultiplication respectively.
Such a set of postulates may be regarded as a distillation of
experience. Centuries of working with numbers and getting
useful results according to the rules of arithmetic - empirically
arrived at - suggested most of the rules embodied in these
precise postulates, but once the suggestions of experience are
understood, the interpretation (here common arithmetic) fur-
nished by experience is deliberately suppressed or forgotten,
and the system defined by the postulates is developed abstractly*
on its own merits, by common logic plus mathematical tact.
Notice in particular IV, which postulates the existence of
negatives. We do not attempt to deduce the existence of nega-
tives from the behaviour of positives. When negative numbers
first appeared in experience, as in debits instead of credits, they,
as numbers, were held in the same abhorrence as 'unnatural"
monstrosities as were later the 'imaginary' numbers V  I,
V  2, etc., arising from the formal solution of equations such
as xz -f 1 = 0, x* -f 2 = 0, etc. If the reader will glance back
at what Gauss did for complex numbers he will appreciate more
fully the complete simplicity of the following partial statement
of Hamilton's original way of stripping 'imaginaries' of their
silly, purely imaginary mystery. This simple tiling was one of
the steps which led Hamilton to his quaternions, although
strictly it has nothing to do with them. It is the method and the